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\begin{center}
\vspace*{1.0cm}
{\LARGE{\bf Fluctuation Operators and Spontaneous Symmetry Breaking}}
\vskip 1.5cm
{\large {\bf Manfred Requardt }}
\vskip 0.5 cm
Institut f\"ur Theoretische Physik \\
Universit\"at G\"ottingen \\
Bunsenstrasse 9 \\
37073 G\"ottingen \quad Germany\\
(E-mail: requardt@theorie.physik.uni-goettingen.de)
\end{center}
\vspace{1 cm}
\begin{abstract}
In the following we develop an in various respects new approach to
this field, which was to a large extent developed by Verbeure et
al. and which may complement their approach, which is largely based
on a non-commutative central limit theorem. In contrast to that we
deal directly with the limits of $l$-point truncated correlation
functions and show that they typically vanish for $l\geq 3$ provided
that the respective scaling exponents of the fluctuation observables
are appropriately chosen. This direct approach is greatly simplified
by the introduction of a smooth version of spatial averaging, which
has a much nicer scaling behavior and the systematic developement of
Fourier space and energy-momentum spectral methods. We both analyze
the regime of normal fluctuations, the various regimes of poor
clustering and the case of spontaneous symmetry breaking or Goldstone
phenomenon.
\end{abstract} \newpage
\setcounter{page}{1}
\section{Introduction}
In the past decade Verbeure and coworkers developed in a series of
papers a beautiful and ingeneous framework to study so-called
macroscopic fluctuation phenomena in systems and various regimes of
quantum statistical mechanics (see the cited literature). The approach
is to a large extent based on a quantum variant of the \tit{central
limit theorem} and is mainly performed in real (i.e. configuration)
space. Among other things, the general goal is it, to study the limit
behavior of correlation functions of so-called \tit{fluctuation
observables}, i.e. appropriately renormalized averages of
microscopic observables, averaged over volumes, $V$, which approach
the whole space, $\R^n$, say. Typically, one arrives, depending on the
type of clustering of the microscopic $l$-point functions, at certain
simple limit algebras as e.g. $CCR$.
We approach the field from a slightly different angle. In a first step
we choose another averaging procedure, which avoids sharp volume
cut-offs and, a fortiori, has a very nice and transparent scaling
behavior. This is then exploited in the following analysis which
systematically develops so-called Fourier-space and energy-momentum
spectral methods of observables and correlation functions. We consider
it to be an advantage that the calculations turn out to be relatively
transparent and lead in a direct way to the desired results.
We first treat the case of \tit{normal fluctuations} and
$L^1$-clustering. We show that all the truncated $l$-point functions
vanish for $l\geq 3$ while they approach a finite, non-trivial limit
for $l=2$. The analysis is done both for the $(k=0)$- and the $(k\neq
0)$-modes. We emphasize that the calculations for net-momentum
different from zero remain also very simple. A variant of the method
is then applied to the case of $L^2$-clustering.
In the second part of the paper we embark on the analysis of
fluctuations in the presence of \tit{spontaneous symmetry breaking
(ssb)}. In a first step we prove some general results in the context
of $ssb$ and the \tit{Goldstone phenomenon}. We then address the
problem of macroscopic fluctuations within this context. Among other
things, we give a general and rigorous proof that the limt
fluctuations are always classical for temperature states (a phenomenon
already observed by Verbeure et al in various simple models). The paper
ends with a treatment of extremely poor clustering, which can be
controlled by a new method we develop in the last section.
To sum up, we think that in our view the two different frameworks seem
to neatly complement each other and should lead to further interesting
results if being combined.
\section{The Scenario of Normal Fluctuations}
The following analysis works for statistical equilibrium states and/or
for vacuum states in quantum field theory. To avoid constant
mentioning of the respective scenario we are actually working in, we
usually treat equilibrium (i.e. KMS-) states, to fix the framework. Now, let
$\Omega$ be the vacuum or equilibrium state (rather its
GNS-representation; usually we work within a concrete Hilbert
space). As an abstract state we denote it by $\omega$. Expectations of
observables are written as
\begin{equation}\langle A\rangle=
\omega(A)=(\Omega,A\Omega)\end{equation}
with $A$ taken from the \tit{local algebra}, $\mcal{A}_0\subset
\mcal{A}$, the latter one being the \tit{quasi-local} norm closure of
$\mcal{A}_0$. We assume $\Omega$ to be \tit{cyclic} with respect to
$\mcal{A}_0$ or $\mcal{A}$. That is, we assume
\begin{equation}\overline{\mcal{A}_0\cdot\Omega}=\mcal{H}\end{equation}
There are certain differences as to the locality properties
of the dynamics between (non-)relativistic statistical mechanics and
relativistic quantum field theory (RQFT). Denoting the dynamics
(acting on the algebra of observables), more properly, the time
evolution, by $\alpha_t$, we have
\begin{ob}In RQFT we usually assume
\begin{equation}\alpha_t:\;\mcal{A}_0\to \mcal{A}_0\end{equation}
while in statistical mechanics (due to weaker locality behavior) we
have in the generic case only
\begin{equation} \alpha_t:\;\mcal{A}\to\mcal{A}\end{equation}
while $\mcal{A}_0$ is usually not left invariant as the
observables will develop infinite but weak tails.
\end{ob}
Furthermore, we assume once for all that our system is in a \tit{pure,
translation invariant phase}, that is $\Omega$ is extremal
translation invariant under the space translations ( which could, as
e.g. in the case of lattice systems, also be a discrete subgroup).
There can of course exist other pure phases at the same external
parameters as e.g. below a phase transition threshold. These
assumptions imply that we can expect certain \tit{cluster properties},
i.e. decay of \tit{correlations} (see e.g. \cite{Ruelle}).
%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Definition of Ordinary Fluctuation Operators}
We begin by defining the \tit{fluctuation operators} in the
\tit{normal} situation as it was done in \cite{Verbeure1}.
We assume, for the time being, $L^1$-clustering for the
two-point-function, that is
\begin{equation}\int|\langle A(x)B\rangle^T|d^nxFrom the above we see that the original hierarchy of $n$-point
functions can be reconstructed from the new hierarchy of truncated
$n$-point functions, which have more transparent cluster properties.
The $L^1$-condition allows us to Fourier transform the
$W^T(x_1,\ldots,x_l)$ and we get from translation invariance:
\begin{multline}const\cdot\int\tilde{W}^T(p_1,\ldots,p_l)\cdot e^{-i\sum
p_ix_i}\prod dp_i=W^T(x_1,\ldots,x_l)
=W^T(x_1-x_2,\ldots,x_{l-1}-x_l)\\= const\int
\hat{W}^T(p_1,p_1+p_2,\ldots,p_1+\cdots
p_{l-1})\cdot\delta(p_1+\cdots p_l)e^{-i\sum p_ix_i}\prod dp_i\\
= const\int \hat{W}^T(q_1,\ldots,q_{l-1})e^{-i\sum_{i=1}^{l-1}
q_iy_i}\prod_{i=1}^{l-1} dq_i
\end{multline}
with
\begin{equation}\label{q} y_i:= x_i-x_{i+1}\;,\;q_i=\sum_{j=1}^i p_j\quad i\leq
(l-1)\end{equation}
The functional determinant $det(\partial q/\partial p)$ is one and we
can regard $\hat{W}^T$ both as a function of the $q_i$'s or the
$p_i$'s. We hence have
\begin{lemma}$\hat{W}^T(p_1,\ldots,p_{l-1})=\hat{W}^T(q_1,\ldots,q_{l-1})$
is, as a Fourier transform of a $L^1$-function a continuous and
bounded function which decreases at infinity in the $q$-variables.
\end{lemma}
%%%%%%%%%%%%%%%%%
\subsection{The $(k=0)$-Modes}
We now study the limit of truncated $l$-point functions with the entries being
fluctuation operators $A^F_R$, more precisely their Fourier
transforms, i.e.
\begin{multline}\langle A^F_R(1)\cdots A^F_R(l)\rangle^T=const\cdot
R^{ln/2}\cdot\int \hat{f}(Rp_1)\cdots \hat{f}(-R[p_1+\cdots
+ p_{l-1}])\cdot\\ \hat{W}^T(p_1,\ldots,p_{l-1})\prod dp_i\\
=R^{ln/2}\cdot R^{-(l-1)n}\cdot\int \hat{f}(p_1')\cdots
\hat{f}(-[p_1'+\cdots+p_{l-1}'])\cdot
\hat{W}^T(p'_1/R,\ldots,p'_{l-1}/R)\prod dp'_i\end{multline}
$\hat{W}$ is continuous and bounded and the $\hat{f}$'s are of rapid
decrease. Hence we can perform the limit $R\to\infty$ under the
integral and get
\begin{ob}The above expression scales as $\sim R^{(2-l)n/2}$. We
hence have that for $l>2$ the above limit is zero, for $l=2$ the
limit is a finite number bounded away from zero in general. In other
words
\begin{equation}\lim_{R\to\infty} \langle
A^F_R(1)\cdots A^F_R(l)\rangle^T=0\;\text{for}\;l>2\end{equation}
and
\begin{equation}\lim_{R\to\infty}\langle
A^F_R(1)\cdots A^F_R(l)\rangle=
\lim_{R\to\infty}\sum_{part}\prod_{\{ij\}}\langle A^F_R(i)A^F_R(j)\rangle\end{equation}
\end{ob}
The relation between the original microscopic system
$(\mcal{A},\omega)$ and the coarse-grained system of fluctuation
operators is a little bit subtle. Note that $\omega_F$ can no longer
be considered as a state or something like that on the original
algebra nor can the fluctuation operators be considered as a
representation of, say, $\mcal{A}$. One aspect of the impending
problems can perhaps best be seen by comparing e.g.
\begin{equation}(A\cdot B)_V^F \neq A_V^F\cdot B_V^F\end{equation}
which pertains also in the limit. That is, in a sense to be defined,
we have
\begin{equation}(A\cdot B)^F\neq A^F\cdot B^F\end{equation}
the same holding in general for all the higher products. This is one
source of non-uniqueness as there is no invariant discrimination
between an observable regarded as a single object to be scaled and as
a product where now each factor has to be scaled. The appropriate
point of view has to be a different one (as has also been emphasized
by Verbeure et al, cf e.g. \cite{Verbeure1}, second ref. p.540f and
private communication).
To begin with, the picture is relatively clear for the intermediate
scales. We have a start system $(\mcal{A},\om)$, labelled by, say,
$V=0$. On every scale $V$ we have a new algebra, $\mcal{A}^F_V$,
(actually a subalgebra of $\mcal{A}$), generated by the observables
$A_V^F\,,\,A\in \mcal{A}$ and including arbitrary finite products
$(A_1\cdots A_n)^F_V$. If we prefer to consider this algebra on scale
$V$ as a new abstract algebra (i.e. forgetting about the underlying
finer algebra $\mcal{A}$), we get also a new, coarse-grained state via
\begin{equation}\om^F_V(A_V^F):=\om(A_V^F)\end{equation}
Remark: A related philosophy was expounded by Buchholz and Verch in
e.g. \cite{Buchholz} within the context of the algebraic analysis of
ultra-vilolet behavior in quantum field theory. \\[0.3cm]
The map
\begin{equation}R_V:\;\mcal{A}\to\mcal{A}^F_V\end{equation}
can be viewed as kind of a \tit{renormalization map}, which does
however \tit{not} preserve the algebraic structure (i.e.the algebras
are in general not \tit{isomorphic}). Furthermore one gets a ``\tit{new}''
dynamics on this algebra by defining
\begin{equation}\alpha^V_t(A_V^F):=(\alpha_t(A))^F_V\end{equation}
(Note however that $\alpha_t$ is assumed to commute with the space
translations, that is, we have $\alpha_t(A^F_V)=(\alpha_t A)^F_V$).
\begin{bem}It may be reasonable to scale the time variable on the lhs
also.
\end{bem}
On the other hand, in order to concretely reconstruct the limit
theory, one can proceed in a slightly different direction.
The above limits of n-point functions define a consistent hierarchy of
new n-point functions which then allow to define a \tit{new} system
living on another state via a so-called \tit{reconstruction theorem}
(for a pendant in quantum field theory see e.g. \cite{Wightman}). Put
differently, we define limit objects, $\{A_i^F\}$, the so-called
fluctuation operators, which live on a new state, $\omega_F$, defined
by the limits:
\begin{equation}\label{limit} \omega_F(A_1^F\cdots A_n^F):=
\lim_{R\to\infty}\langle A^F_{1,R}\cdots
A^F_{n,R}\rangle=\sum_{part}\prod_{\{ij\}}\omega_F(A^F_i\cdot
A^F_j)\end{equation}
Note however that the so-called \tit{Gelfand-ideal}, $I_F$, is large,
that is, there are a lot of elements of $\mcal{A}$ which are mapped to
zero by this limit with
\begin{equation}I_F:=\{A\,;\,\omega_F((A^F)^*\cdot A^F)=0\}\end{equation}
This is of course typical for such kind of \tit{mean-values}, as
e.g. all space-translates of $A$ yield the same limit
element. Shifting one of the observables in the above $l$-point
functions by, say, $a_i$ yields an extra factor $e^{ip_ia_i}$ in the
Fourier transform which after the above coordinate tranformation goes
over into $e^{ip_i'/R\cdot a_i}$ which goes to one. Summing up we have
\begin{obdef} Via (\ref{limit}) we get a new system, the {\em algebra
of fluctuation operators}, $\mcal{A}_F$, which lives on the state
$\omega_F$. The well-known GNS-construction (see
e.g. \cite{Bratteli1}) yields the corresponding Hilbert-space
representation with
\begin{equation}\omega_F(A_1^F\cdots A_n^F)=(\Om_F,A_1^F\cdots
A_n^F\Om_F)\end{equation}
(where, by abuse of notation, we do not discriminate between operators
and their equivalence classes on the rhs).
As all the $n$-point functions decay into a product of $2$-point
functions all the commutators are $c$-numbers:
\begin{equation}[A^F,B^F]=\omega_F([A^F,B^F])\end{equation}
The system of fluctuation operators is a {\em quasi-free} system
(cf. \cite{ Bratteli2})
\end{obdef}
Taking now self-adjoint elements one can, as in \cite{Verbeure1},
represent the new system as a representation of the $CCR$ over the real vector space of
s.a. operators. Our scalar product, given via the hierarchy of
$n$-point functions, can be split in the following way.
\begin{equation}(A^F\Om_F,B^F\Om_F)=Re\;(\ldots)+i\,Im\;(\ldots)=:s_F(A^F,B^F)+(i/2)\sigma_F(A^F,B^F)\end{equation}
\begin{equation}\om_F([A^F,B^F])=\sigma_F(A^F,B^F)\end{equation}
where $\sigma_F$ defines a \tit{symplectic form}. The
\tit{Weyl-operators}, $e^{iA^F}$ with $A^F$ s.a., fulfill the
$CCR$-relations, e.g.
\begin{equation}\om_F(e^{iA^F})=e^{-1/2s_F(A^F,A^F)}\end{equation}
\begin{equation} e^{iA^F}\cdot e^{iB^F}=e^{i(A^F+B^F)}\cdot e^{-i/2\sigma_F(A^F,B^F)}\end{equation}
Remark: In our context the first equality can e.g. be verified as
follows: Only the even, i.e. $2n$-point functions are different from
zero. On the lhs we hence have
\begin{equation}\label{exp} \om_F(e^{iA^F})=\sum
(-1)^n/(2n)!\cdot\om_F([A^F]^{2n})\end{equation}
The only non-trivial task consists in counting the number of
partitions of an $2n$-set into $2$-sets. As we could not find a
\tit{natural} i.e. \tit{combinatorial} proof of the, however,
well-known result, we give here our own one.
\begin{lemma}The number of partitions of an $2n$-set into $2$-sets is
$(2n)!/2^n\cdot n!$.
\end{lemma}
Proof: The number of all the \tit{ordered} $2n$-tuples is
$(2n)!$. Each of these $2n$-tuples of elements, $e_{i_1}\ldots
e_{i_{2n}}$, we partition as
$(e_{i_1}e_{i_2})\ldots(e_{i_{2n-1}}e_{i_{2n}})$. By this
construction we create repetitions of ordered partitions. There are
$n!$ repetitions of such groupings in $2$-sets and each \tit{unordered}
$2$-set occurs twice as $(e_{i_1}e_{i_2})$ or vice versa. Hence we
have to divide by $2^n\cdot n!$ and get the desired
result.\hfill$\Box$\\[0.5cm]
In (\ref{exp}) we now get for $A^F$ s.a. on the rhs
\begin{equation}\sum_n 1/n!(-1/2\cdot
\om_F(A^FA^F))^n=e^{-1/2s_F(A^FA^F)}\hfill\Box\end{equation}
The above general cluster result of the limit $n$-point functions make
the study of the limit time evolution relatively straightforward. It
suffices, in a first step, to study the $2$-point functions. We define
time evolution in the limit theory by
\begin{equation}\om_F(A^F(t')\cdot B^F(t)):=\lim\om(A_V^F(t')\cdot
B_V^F(t))=\lim\om(A(t')_V^F\cdot B(t)_V^F)\end{equation}
On the limiting GNS-Hilbert space constructed above we now get a
bounded \tit{sesquilinear form} $(x,y(t))$ which, by standard results,
yields a bounded operator $U^F(t)$ implementing the time
evolution. Here we use that the limit n-point functions are products
of $2$-point functions. Furthermore
we see by the same reasoning as above that
\begin{equation}(U_t^Fx,U_t^Fy)=\om_F(\ldots)=\lim\om(\ldots)=(x,y)\end{equation}
\begin{conclusion}The above construction yields a strongly continuous
unitary time evolution on the limiting GNS-Hilbert space.
\end{conclusion}
Another point worth to mention is the question of the non-triviality
of the commutators
\begin{equation}[A^F,B^F]=\om_F([A^F,B^F])\end{equation}
In principle it could happen that all the expectation values on the
rhs vanish. In that case the limit algebra would be \tit{abelian} and
the fluctuations \tit{classical}. In a more general context
(cf. e.g. \cite{Buchholz}) this problem is more complicated. In our
context this question can however be answered in a rather
straightforward way. We have
\begin{equation}\lim_V\om([A^F_V,B^F_V])=\lim_V\om([A_V,V^{-1}\cdot
B_V])\end{equation}
For $A,B\in\mcal{A}_0$, i.e. local, the rhs equals
\begin{equation}\lim_V\om([A_V,B])\end{equation}
We know candidates which lead to a vanishing of the limit for all
$B\in\mcal{A}_0$. For $A$ chosen s.a. these are the generators of
\tit{conserved symmetries}, written
\begin{equation}Q:=\int A(x)d^nx\end{equation}
Usually they are assumed to commute with the time evolution, written
$Q(t)=Q$, hence the above limit would also be zero on the full
quasi-local algebra.
This situation, more specifically the case of \tit{spontaneous
symmetry breaking (ssb)} and \tit{Goldstone phenomenon}, will be
dealt with in more detail in section \ref{Goldstone}. In any case,
as conserved symmetries are usually not so numerous, we may presume
that, in the generic case, not all of these commutators will be zero.
For $A,B$ not necessary strictly local our above more general
formalism is useful. Calling
\begin{equation}\om(A(x)B)=F_{AB}(x)\quad,\quad\om(BA(x))=G_{AB}(x)\end{equation}
we conclude:
\begin{multline}0=[A^F,B^F]=\lim_R R^n\cdot\int
|\hat{f}(Rp)|^2(\hat{F}_{AB}(p)-\hat{G}_{AB}(p))d^np\\
=\lim_R\int |\hat{f}(p)|^2(\hat{F}_{AB}(p/R)-\hat{G}_{AB}(p/R))d^np\\
= (\hat{F}_{AB}(0)-\hat{G}_{AB}(0))\cdot\int|\hat{f}(p)|^2d^np
\end{multline}
due to the theorem of dominated convergence (note we are in the
$L^1$-situation).
\begin{conclusion}
\begin{equation}[A^F,B^F]=0\Leftrightarrow
\hat{F}_{AB}(0)=\hat{G}_{AB}(0)\end{equation}
hence
\begin{equation}\int F_{AB}(x)d^nx=\int G_{AB}(x)d^nx\end{equation}
or
\begin{equation}\int (\Om,[A(x),B]\Om)d^nx=0\end{equation}
as above.
\end{conclusion}
%%%%%%%%%%%
\subsection{The $(k\neq 0)$-Modes}
Up to now only the $(k=0)$-modes of fluctuation operators,
i.e. $\lim_V V^{-n/2}\int_V A(x)d^nx$, have been studied. For various
reasons it is useful to have corresponding formulas at hand for
fluctuation observables containing a certain net-momentum. This
problem was studied by Verbeure et al in e.g. \cite{k-Mode} and the
results applied in e.g. \cite{Boson} in the analysis of \tit{Goldstone
modes}. In the original (real-space) approach the (ingeneous)
calculations turned out to be far from being simple. This is another
case in point to demonstrate the merits of our Fourier space scaling
methods.
Instead of the original scaling operators, $A_V^F$ or $A_R^F$, we now
study their $k\neq 0$-variants, $A_R^F(k)$. To fix the notation we
begin with a defintion.
\begin{defi}
\begin{equation}\hat{A}(k):=(2\pi)^{-n/2}\int e^{ikx}A(x)d^nx\end{equation}
(with the convention $\hat{f}(k)=(2\pi)^{-n/2}\int e^{-ikx}f(x)d^nx$)\\
is an operator-valued distribution.
\end{defi}
Remark: For a systematic use of such energy-momentum techniques in
quantum statistical mechanics see e.g. \cite{Quasi} where also some
more mathematical background is
provided.\\[0.3cm]
Integrating now over, say, $e^{iqx}\cdot f_R(x)$, we get the $q$-mode
fluctuation operators.
\begin{multline}A_R^F(q):=R^{-n/2}\int
A(x)e^{iqx}f_R(x)d^nx=R^{n/2}\int \hat{A}(k+q)\hat{f}(Rk)d^nk\\
=R^{n/2}\int \hat{A}(k)\hat{f}(R(k-q))d^nk\end{multline}
We can now proceed in exactly the same way as above in the case of the
zero-mode analysis and calculate the truncated $l$-point functions
$\langle A_R^F(1,q_1)\cdots A_R^F(l,q_l)\rangle^T$ (where the indices
$1$ to $l$ denote different observables). The only thing
that changes are the test functions, i.e. $f_R(x)\to e^{iq_kx}\cdot
f_R(x) $. We arrive at the conclusion:
\begin{conclusion}[$q$-Mode Fluctuation Operators]\hfill\\
In the case of $L^1$-clustering all truncated correlation functions
vanish for $l\geq 3$ and the $l$-point functions are again sums of
products of $2$-point functions. The concrete form of the
limit-$2$-point functions is calculated below.
\end{conclusion}
It is perhaps worthwhile to calculate the limt-$2$-point functions
explicitly. We have
\begin{multline}\langle A_R^F(q_1)\cdot B_R^F(q_2)\rangle^T =
R^n\int\langle \hat{A}(k_1+q_1)\hat{B}(k_2+q_2)\rangle^T\cdot
\delta(k_1+q_1+k_2+q_2)\cdot \hat{f}(Rk_1)\hat{f}(Rk_2)dk_1dk_2\\
=R^n\int\ \langle
\hat{A}(k_1+q_1)\hat{B}(-(k_1+q_1))\rangle^T\cdot\hat{f}(Rk_1)\hat{f}(-R(k_1+q_1+q_2))dk_1\\
=R^n\int\langle
\hat{A}(k)\hat{B}(-k)\rangle^T\cdot\hat{f}(R(k-q_1))\hat{f}(-R(k+q_2))dk\end{multline}
With $k':=R(k-q_1)$ we arrive at
\begin{equation}\int \hat{W}^T(k'/R+q_1)\cdot
\hat{f}(k')\hat{f}(-k'-R(q_1+q_2))dk'\end{equation}
By assumption $\hat{W}^T$ is in $L^1$, $\hat{f}$ is of rapid decrease,
so the limit can again be carried out under the integral and we get
\begin{conclusion}For $q_1+q_2\neq 0$ we have
\begin{equation}\lim_R\langle A_R^F(q_1)\cdot B_R^F(q_2)\rangle^T
=0\end{equation}
For $q=q_1=-q_2$ we have on the other side
\begin{equation}\lim_R\langle A_R^F(q)\cdot
B_R^F(-q)\rangle^T=\hat{W}^T(q)\cdot\int
\hat{f}(k)\hat{f}(-k)dk\end{equation}
In other words, the limit tests the spectral momentum of the two-point
function.
\end{conclusion}
%%%%%%%%%%%%%%%%%%%%
\section{The Case of $L^2$-Clustering}
Before we embark on an investigation of the situation in the regime
where phase transitions, vacuum degeneracy and/or spontaneous symmetry
breaking (ssb) prevail, we briefly address the case where the
clustering is weaker than $L^1$ but still $L^2$, say. It is a merit of
the above Fourier-space approach that such case can easily be handled.
\begin{assumption}The truncated $l$-point functions display
$L^2$-clustering in the difference variables.
\end{assumption}
Now we cannot conclude that the F.tr. is bounded and continuous, but
we know it is again an $L^2$-function.
We repeat the first steps of the above calculation with, however,
another \tit{scaling exponent}, $\alpha$, which we leave open for the
moment.
\begin{defi}In the following we define
\begin{equation}A_R^F:=R^{-\alpha}\cdot\int
A(x)f_R(x)d^nx\end{equation}
\end{defi}
We get
\begin{equation}\langle A_R^F(1)\cdots A_R^F(l)\rangle^T=const\cdot
R^{l(n-\alpha)}\cdot\int
\hat{f}(Rp_1)\cdots\hat{f}(-Rq_{l-1})\cdot
\hat{W}^T(q_1,\ldots,q_{l-1})\prod dq_i
\end{equation}
where the $\{p_i\}$ are linear functions of the $\{q_i\}$ as described
above. We now conclude
\begin{multline}\label{L^2}|lhs|\leq const\cdot
R^{l(n-\alpha)}\left[\int(\hat{f}(Rp_1)\cdots
\hat{f}(-Rq_{l-1}))^2\prod
dq_i\right]^{1/2}\\\cdot\left[\int(\hat{W}^T(q_1,\ldots,q_{l-1}))^2\prod dq_i\right]^{1/2}
\end{multline}
In the first integral on the rhs we make again a variable
transformation from $q_i$ to $q'_i:=Rq_i$, yielding an overall scaling
factor
\begin{equation}R^{l(n-\alpha)}\cdot R^{-(l-1)n/2}\end{equation}
We again want the limits of the $2$-point functions to be both finite and
non-trivial, i.e. different from zero in general.
\begin{ob} To make the rhs of (\ref{L^2}) finite in the limit for $l=2$ the {\em
maximal} $\alpha$ to choose is
\begin{equation}3n-4\alpha=0\quad\text{i.e.}\quad
\alpha=(3/4)n\end{equation}
For a general $l$ this leads to the scaling exponent $(n-(1/2)l\cdot n)/2$,
which is negative for $l\geq 3$. Hence, all higher truncated $l$-point
functions vanish in the limit.
\end{ob}
To guarantee, however, that the result is really non-trivial we have
to analyze the situation in more detail as the above estimate is only
an inequality. In the case of $L^1$-clustering $\alpha=n/2$ was
appropriate. The largest value which can occur in the $L^2$-case is
the above maximal $\alpha=(3/4)n$. If we want to avoid that the
$2$-point functions vanish in the limit we have to choose in the
$L^2$-case
\begin{equation}(1/2)n(2/3)n$, we get
a negative scaling exponent for $l\geq 3$ as
\begin{equation}n-(1/3)ln\leq 0\quad\text{for}\quad l\geq
3\end{equation}
For $\alpha=2/3$ the $3$-point functions are finite.
\end{ob}
\begin{bem}One would get corresponding relations for smaller $\alpha$
but higher correlation functions, beginning from a certain order,
$l_0(\alpha)$ say. On the other hand, one cannot guarantee the apriori
existence of the $l$-point functions for $2n\quad\text{and}\quad
\alpha>(1/2)n\end{equation}
and $\alpha$ being so chosen that the $2$-point functions are
non-trivial.
\end{bem}
%%%%%%%%%%%%%%%%%%%%%
\section{Spontaneous Symmetry Breaking (SSB) and the Goldstone Phenomenon}
\label{Goldstone}
\subsection{General Remarks}
Before we embark on the discussion of fluctuation operators in the
regime of vacuum-, ground-, equilibrium-state degeneracy, we want, in
order to set the stage, to briefly comment on the (rigorous)
implementation of $ssb$ in the various fields with particular emphasis
on (quantum) statistical mechanics, i.e. condensed matter physics. As
this topic has however been much discussed in the past from various
points of views, we do not intend to give an exaustive commentary. We
only mention some earlier work being of relevance for our argumentation
and sketch the general framework.
We assume that our state, $\om$ or $\Om$, is (non-)invariant under some
automorphism group of $\mcal{A}_0$ or $\mcal{A}$. Furthermore, and
this is important (while frequently not clearly stated), we assume the
time evolution to commute with the automorphism group.
\begin{defi}$\alpha_g$ is called a symmetry group if
\begin{equation}\alpha_g\cdot\alpha_t=\alpha_t\cdot\alpha_g\end{equation}
\end{defi}
\begin{obdef}If
\begin{equation}(\Om,\alpha_g(A)\Om)=(\Om,A\Om)\end{equation}
for all $A\in\mcal{A}$, the symmetry is called {\em conserved} and can
be implemented by a unitary group of operators in the representation
space
\begin{equation}\alpha_g(A)\;\to\;U(g)AU(g^{-1})\end{equation}
If
\begin{equation}(\Om,\alpha_g(A)\Om)\neq(\Om,A\Om)\end{equation}
for some $A$, $A$ the {\em symmetry-breaking observable}, the symmetry
is called {\em spontaneously broken} while it still commutes with the
time evolution (i.e. formally: with the Hamiltonian, modulo {\em
boundary terms} due to {\em long-range correlations}).
\end{obdef}
In most cases the (continuous) symmetry group derives from a clearly
identifiable \tit{generator} (we restrict ourselves, for convenience,
to one-parameter groups) which is built from a local operator density,
i.e.
\begin{equation}U(s)=e^{isQ}\quad,Q(t)=\int
q(x,t)d^nx\;,\;Q(t)=Q(0):=Q\end{equation}
Note that there are a lot of technical subtleties lurking behind these
operator identities, all of which we cannot mention in the following
(for more details and references see e.g. \cite{ssb}. A nice review is
\cite{Wre} where many of the widely scattered results have been compiled
).
\begin{bem}Frequently the generator density is the zero-component of a
{\em conserved current}. Formally the conservation law encodes the
time-independence of the global charge, $Q$. Furthermore, for
convenience, we assume the symmetry to commute with the space
translations, i.e. $U(x)QU(-x)=Q$. This is in fact frequently the
case and simplifies certain calculations.
\end{bem}
The most crucial consequence is that in case the symmetry is
spontaneously broken some of the above relations do only hold in a
formal or algebraic sense. More specifically:
\begin{ob}If $\alpha_g$ is spontaneously broken the global generator $Q$ does only exist
in a formal sense as a limit
\begin{equation}Q=\lim_V Q_V\quad,\quad Q_V:=\int_V
q(x)d^nx\end{equation}
We have
\begin{equation}ssb\Leftrightarrow\lim_V(\Om,[Q_V,A]\Om)\neq 0
\label{Q}\end{equation}
for some $A\in\mcal{A}$ and $Q$ is only definable as a {\em nasty}
operator (see below).
\end{ob}
In the following we will take (\ref{Q}) as the defining relation of
$ssb$ (that this is correct is shown in the above mentioned
literature).
The notion of $ssb$ is closely connected with another famous
phenomenon, the so-called \tit{Goldstone-phenomenon}. While there
exists a clear picture in, say, \tit{relativistic quantum field
theory}, the corresponding picture is a little bit blurred in the
non-relativistic regime. In the relativistic context we have sharp
\tit{zero-mass Goldstone-modes}, i.e. true particles due to
relativistic covariance. On the other hand, in e.g. condensed matter
physics or statistical mechanics the situation is less generic. In
general we do no longer have sharp excitation modes; we have rather
to expect excitation modes having a \tit{finite lifetime} for momentum
different from zero but becoming infinitely sharply peaked for
momentum $k\to 0$. The proper view is it to analyze these
excitation branches in the \tit{full} Fourier-space of
\tit{energy-momentum} as have been done in ref. four of \cite{ssb} and
earlier in the author's doctoral thesis, the principal object being
the spectral-resolution of the $2$-point correlation functions (in a
neighborhood of $(E,k)=(0,0)$). $SSB$ or the Goldstone phenomenon
manifests itself by a singular contribution in the spectral measure.
It turns out that in the non-relativistic regime the concrete
structure of the Goldstone mode depends usually on the details of the
microscopic interactions (that means both the so-called
energy-momentum dispersion-law which can be, to give an example,
quadratic or linear near $k=0$ in the case of magnons or phonons, say,
and the $k$-dependent width of the branch). This led to the desire to
characterize the presence of a Goldstone phenomenon by a simple (if
qualitative) property. Sometimes one finds in the literature the
saying that the Goldstone phenomenon consists of the vanishing of a
\tit{mass-gap} above the ground state. But this statement is in some
sense frequently empty. From \cite{Swieca} we know e.g. that a
\tit{short-ranged} \tit{Galilei-covariant} theory, with a
non-vanishing particle density, cannot have a mass-gap due to
\tit{phonon-excitations} which signal the trivial breaking of the
Galilei-boosts.
\begin{bem}Models like the famous BCS-model (having a gap) are no case
in point as they are implicitly breaking Galilei-invariance as do
all such {\em mean-field-models}. This becomes apparent when
analyzing the interaction part of the corresponding Hamiltonian. The
complete fermion- or boson-liquid is, on the other side, again
Galilei-invariant, hence has no mass-gap, but may, of course, still
display e.g. {\em superfluidity}.
\end{bem}
In the next subsection we will provide a, as we think, more satisfying
and completely general characterization of the Goldstone phenomenon
which is independent of the details of the model under discussion.
%%%%%%%%%%%%%%%%
\subsection{Some Rigorous Results for the Symmetry Generator in the
Presence of SSB}
After the above introductory remarks we want to prove a couple of
rigorous results which characterize to some extent the presence of
$ssb$ in the (non-)relativistic regime. The main observation is that
the symmetry generator is no longer defined as a nice operator in the
representation (Hilbert- or $GNS$-) space when $ssb$ is present.
Let us work, for simplicity, in the context of \tit{temperature
states}. This has the advantage that $\Om$ is separating, i.e.
\begin{equation}A\Om=B\Om\Rightarrow A=B\end{equation}
The first task is to give $Q:=\lim_V Q_V$ a rigorous meaning. The
standard procedure (see the above mentioned literature) is to define
$Q$ via:
\begin{equation}QA\Om:=\lim_V [Q_V,A]\Om\quad,\quad
Q\Om:=0\end{equation}
for e.g. $A\in\mcal{A}_0$. For $V$ sufficiently large, the commutator
on the rhs becomes independent of $V$, hence there is a chance to get
a well-defined $Q$ (at least on a dense set of vectors) as on the lhs
we have by \tit{separability}
\begin{equation}A\Om=B\Om\Rightarrow A=B\Rightarrow
[Q_V,A-B]=0\end{equation}
For $A\in\mcal{A}$ one has to employ cluster properties.
\begin{bem}We have already seen above that, while such a $Q$ may
exist, the corresponding $\|Q_V\Om\|$ will nevertheless diverge for
$V\to\R^n$! This shows that the connection between the global
generator and its local approximations is not that simple. The best
one can expect in the case of \tit{symmetry conservation} is a weak
convergence on a dense set
\begin{equation}(B\Om,QA\Om)=\lim_V(B\Om,Q_V\Om)\end{equation}
but, due to the above divergence of $\|Q_V\Om\|$, we cannot even have
weak convergence on the full Hilbert-space. (For more details see the
above cited literature; in particular \cite{ssb}, third ref., where
the various possibilities in the respective fields have been compared)
\end{bem}
We see from the above that $Q$ can be defined as a densely defined
operator but usually we do want more. A conserved continuous symmetry
is given by a s.a. generator. Let us see under what conditions the
above $Q$ is at least \tit{symmetric} provided that the $Q_V$ are
symmetric. We assume the symmetry to be conserved, i.e.
\begin{equation}\lim_V(\Om,[Q_V,A]\Om)=0\quad\text{for all}\quad
A\in\mcal{A}\end{equation}
We then have
\begin{multline}(B\Om,QA\Om)=\lim_V(B\Om,[Q_V,A]\Om)\\
=\lim_V\left(([Q_V,B]\Om,A\Om)+(Q_v\Om,B^*A\Om)-(A^*B\Om,Q_V\Om)\right)
\end{multline}
\begin{ob}$Q$ is symmetric if $\lim_V(A\Om,Q_V\Om)=0$ for all
$A\in\mcal{A}_0$. Under the same proviso it follows
\begin{equation}(B\Om,QA\Om)=\lim_V(B\Om,Q_V\Om)\end{equation}
\end{ob}
What is the situation if the symmetry is spontaneously broken? For
convenience we replace again the sharp volume-integration by our
smooth one, i.e.
\begin{equation}Q_V\to Q_R:=\int q(x)f_R(x)d^nx\end{equation}
We know that there exists a symmetry-breaking observable $A$ s.t.
\begin{equation}\lim_R(\Om,[Q_R,A]\Om)\neq 0\Rightarrow
QA\Om=\lim_R[Q_R,A]\Om\neq 0\end{equation}
Due to the assumed translation invariance, i.e.
\begin{equation}U(a)QU(-a)=Q\quad\text{or, what is the same,}\quad U(a)q(x)U(-a)=q(x+a)\end{equation}
we have
\begin{equation}(\Om,QA\Om)=(\Om,Q\cdot V^{-1}A_V\Om)\end{equation}
and
\begin{equation}Q\cdot V^{-1}A_V\Om=V^{-1}\int_V
U(x)d^nx\cdot QA\Om\end{equation}
$U(x)$ the unitary representation of the translations.
\begin{lemma}
\begin{equation}s-\lim_V V^{-1}\int_V U(x)d^nx=P_{\Om}\end{equation}
$P_{\Om}$ the projector on the (in our case) unique vacuum-,ground-,
equilibrium-state.
\end{lemma}
Proof: The result is well-known (see e.g. \cite{Ruelle}). We give
however a very short and slightly different proof using our smooth volume integration.
With $V_R:=\int f_R(x)d^nx$, a \tit{spectral resolution} yields
\begin{equation}V_R^{-1}\cdot\int U(x)f_R(x)d^nx=const\cdot(\int
f(x)d^nx)^{-1}\cdot\int(\hat{f})(Rp)dE_p\end{equation}
Applied to a vector $\psi$ we can now employ Lebesgue's theorem of
dominated convergence and get
\begin{equation}\lim_V
V^{-1}\int
U(x)d^nx\cdot\psi=(\hat{f}(0))^{-1}\cdot\hat{f}(0)P_{\Om}\psi=P_{\Om}\psi\quad\Box
\end{equation}
This yields
\begin{equation}0\neq P_{\Om}QA\Om=\lim_V Q\cdot
V^{-1}A_V\Om\end{equation}
On the other hand
\begin{equation}\lim_V\|V^{-1}A_V\Om\|=\|P_{\Om}A\Om\|=0\end{equation}
by an analogous reasoning (note that $(\Om,A\Om)=0$).
We have now a sequence of vectors, $V^{-1}A_V\Om$, converging to zero
in norm while $Q\cdot V^{-1}A_V\Om$ converges to $P_{\Om}QA\Om\neq 0$.
\begin{conclusion}[Goldstone Theorem]In case of $ssb$ and a
separating $\Om$, $Q$ can still be defined as an operator which is,
however, not closable, hence, a fortiori, not symmetric (note that
symmetric operators are closable). Furthermore this seemingly
abstract result is encoded in the concrete physical property
exhibited in the above formulas. In our view they express the
content of the Goldstone phenomenon in the most general and model
independent way. They show that $Q$ induces transitions from a
singular part of the continuous spectrum, passing through $(E,p)=(0,0)$, to the extremal invariant
state $\Om$ (see the remarks below). On the other side, a conserved
symmetry implies
\begin{equation}Q\Om=0\;,\; P_{\Om}[Q,A]\Om=0\Rightarrow
P_{\Om}QA\Om=0\end{equation}
\end{conclusion}
We show now that the above result really contains the original
Goldstone phenomenon. Let us e.g. assume that we have the above result
and, on the other side, a gap in the energy spectrum above the state
$\Om$. We emphasized above that an important ingredient of the notion
of $ssb$ is the time independence of, say, the above expression. We
employ again the spectral resolution of operators with respect to
energy-momentum. We hence have
\begin{equation}0\neq c=P_{\Om}Q\int
\hat{A}(k,E)e^{-itE}dkdE\Om\end{equation}
with $c$ being independent of $t$. We choose a real testfunction
$g(t)$ with $\int g(t)dt=1$. This yields
\begin{equation}0\neq c=P_{\Om}Q\int A(t)\cdot g(t)dt\Om=
P_{\Om}Q\int\hat{A}(E)\hat{g}(E)dE\Om\end{equation}
If there is a gap above zero we may choose the support of $\hat{g}$
so that
\begin{equation}supp(\hat{g})\cap supp(H)=0\end{equation}
Since, by assumption, $P_{\Om}$ has been extracted in the
energy-support of $A$, we get the result $c=0$, that is, no symmetry breaking.
But we can infer more about the nature of the energy-momentum spectrum
near $(0,0)$. We see that $P_{\Om}QA(g(t))\Om$ depends only on the
value of $\hat{g}(E)$ in $E=0$, which is one in our case, but not on
the shape of $g$. We hence can infer
\begin{conclusion}$\hat{A}(E)\Om$ contains a characteristic singular
contribution of $\delta(E)$-type.
\end{conclusion}
This can be seen by writing the above expression as
\begin{equation}0\neq c=(Q^*\Om,\int A(E)e^{-iEt}dE\Om)\end{equation}
and the time-independence of the lhs. This sharp excitation extends
into the full energy-momentum plane in form of a (usually) smeared
excitation branch (having a finite $k$-dependent life-time). For the
regime of temperature states the situation was analyzed in some detail
in the fourth reference of \cite{ssb} and already in the authors
doctoral thesis. We see from the above that a similar situation
prevails in the more general case of a separable $\Om$ and,
analogously, for ground-state models where $Q$ can be defined in the
above way. Even if the above $Q$ is not definable as a non-closable
limit operator we arrive at a similar result by exploiting the
limit-expectation values instead of the strong vector- or operator
limits, but we do not want to dwell more into the corresponding
details in this paper.
%%%%%%%%%%%%%%
\section{The Canonical (Goldstone) Pair in the Presence of $SSB$}
We remarked above that $ssb$ is characterized by the non-vanishing
(but time-independence) of the following commutator limit
\begin{equation} \label{sym} 0\neq c=\lim_V(\Om,[Q_V,A(t)]\Om)\end{equation}
To fix the notation: usually a pure phase is characterized by the
non-vanishing of a so-called \tit{order parameter} in the presence of
$ssb$. This is an observable, $B$ say, with
\begin{equation}
(\Om,B\Om)=\begin{cases}
c\neq 0 & \text{in the broken phase}\\
0 & \text{in the conserved phase (above $T_c$, say)}
\end{cases}
\end{equation}
>From (\ref{sym}) we see that as order parameter we have to choose
\begin{equation}B:=\lim_V [Q_V,A]\end{equation}
while $A$ is the symmetry breaking observable.
\begin{bsp}In the Heisenberg-ferromagnet with spontaneous
magnetization in, say, the $z$-direction the order parameter is
$S_z$ or $\langle S_z\rangle$. As generator of the broken symmetry
one may take $\sum S_x$ and as symmetry breaking obsrvable
e.g. $S_y$.
\end{bsp}
We have seen that we can write
\begin{equation}0\neq c=\lim_V (\Om,[Q_V,A]\Om)=\lim_V
(\Om,[Q_V,V^{-1}A_V]\Om)=(\Om,[Q_R,V_R^{-1}A_R]\Om)\end{equation}
where
\begin{equation}Q_R:=\int q(x)f_R(x)d^nx\;,\;A_R:=\int_{S_R}
A(x)d^nx\end{equation}
with $V_R$ the volume of the sphere, $S_R$, with radius $R$.
We can now split the scaling exponent among the two observables (the
volume of the unit sphere being absorbed in the constant on the rhs).
\begin{equation}0\neq const = \lim_R (\Om,[R^{-\alpha}Q_R,R^{-(n-\alpha)}A_R]\Om)\end{equation}
This form of scaling may yield something reasonable if the scaling
exponents can be so adjusted that also
\begin{equation}(\Om,R^{-\alpha}Q_RR^{-\alpha}Q\Om)\;\text{and}\;(\Om,R^{-(n-\alpha)}AR^{-(n-\alpha)}A\Om)\end{equation}
remain finite in this limit.
In general it does not seem to be easy to get both rigorous and general
estimates on the scaling behavior of these quantities. Fortunately, in
the case of temperature (KMS) states, such estimates are available. In
\cite{Perez} to \cite{Requardt-KMS} the following special
(\tit{real-space}-) version of the \tit{Bogoliubov-Inequality} has
been proved and employed for the observables $Q_R$ and $V_R^{-1}A_R$:
\begin{equation}|\langle[Q_R,V_R^{-1}A_R]\rangle|^2\leq\langle
V_R^{-1}A_RV_R^{-1}A_R\rangle\cdot\langle[Q_R,[Q_R,H]]\rangle\end{equation}
The delicate term is the double commutator on the rhs. If $Q$ is
spontaneously broken boundary terms will survive in the commutator of
$Q_R$ and the Hamiltonian, $H$, when taking the limit $R\to\infty$, while in a formal sense they
commute. The double commutator saves us two powers of $R$, so to
say. That is we arrive after some cumbersome manipulations at
\begin{equation}\langle[Q_R,[Q_R,H]]\rangle\sim
R^{(n-2)}\;\text{for}\;R\to\infty\end{equation}
\begin{equation}\langle V_R^{-1}A_RV_R^{-1}A_R\rangle\gtrsim
R^{(2-n)}\;\text{for}\;R\to\infty\end{equation}
as the limit on the lhs is a constant different from zero in the case
of $ssb$.
\begin{conclusion}For temperature states we have for the symmetry
breaking observable
\begin{equation}\langle A_RA_R\rangle\gtrsim
R^{(n+2)}\end{equation}
That is, compared with the ordinary, normal scaling behavior ($\sim
R^n$), the divergence is worse. From this one gets for the decay of
the two-point correlation function itself:
\begin{equation}|\langle A(x)A\rangle|\gtrsim R^{(n-2)}\end{equation}
\end{conclusion}
Putting all the pieces together we now have to make the following
identification:
\begin{equation}n-\alpha\geq (n+2)/2\;\Rightarrow\;\alpha\leq
(n-2)/2\end{equation}
in order that the limit commutator is non-trivial, i.e. non-classical.
On the other hand, the divergence behavior of $\langle Q_RQ_R\rangle$
can frequently be inferred either from covariance properties (as in
relativistic quantum field theory; see e.g. the third reference in
\cite{Requ2}) or from an analysis of the spectral behavior in concrete
(non-relativistic) models.
\begin{ob}[Canonical Pair] For a covariant four-current in
relativistic quantum field theory the two-point function in Fourier
space contains a prefactor $\sim p^2$ which yields (after some
calculations) an $\alpha=1/2$ (for space dimension, $n=3$). On the
other side, if we do not have such nice covariance properties the
divergence of $\langle Q_RQ_R\rangle$ is generically much worser
than $\sim R$ (in three dimensions). This holds, in particular, for
the above temperature states. It follows that for temperature states
we cannot find a critical exponent $\alpha$ so that both the
auto-correlations remain finite in the limit and the commutator
non-trivial. That is, for temperature states the limit fluctuations
are classical (an observation already made by Verbeure et al for
special models, see e.g. \cite{Boson}).
\end{ob}
The situation is less clear for ground state models, i.e. the
temperature-zero case. For one, we do not have something like the
Bogoliubov inequality which shows that it is the autocorrelation of $
A_R$ which is ill-behaved. For another, in temperature states, as was shown in e.g. the
fourth reference of \cite{ssb}by the author, the spectral weight has
to become infinite along the Goldstone excitation branch in a specific
way (which is governed by the dispersion law of the Goldstone mode)
for energy-momentum approaching zero. This sort of singularity is
mainly responsible for the poor decay of the respective
auto-correlation function. This phenomenon is generally absent in the case of
ground states as has also been shown for certain Bose-gas models in
\cite{Boson}. It should be worthwhile to investigate these problems in
greater generality. Note in particular that a variety of aspects may
depend on the precise shape of the Goldstone mode near energy-momentum
equal to $(0,0)$ as was shown in the above mentioned paper of the
author or in the unpublished doctoral thesis.
%%%%%%%%%%%%%%
\section{The Case of SSB or Very Poor Decay of Correlations}
In the preceding sections we studied the case of $L^1$- or
$L^2$-clustering. In this last section we want to briefly show how we
can proceed in the case of extremely poor clustering. We want
however, for the sake of brevity and in order to better illustrate the
method, to concentrate on the simpler case of a uniformly poor decay
of all the correlation functions we are discussing. This is of course
not always the case but the scheme can be easily generalized.
We hence assume that the truncated $l$-point functions cluster weaker
than $L^2$ or $L^1$, say, in the difference variables,
$y_i:=x_{i+1}-x_i$, (see section 3.1). \\[0.3cm]
Remark: The following reasoning works both in the case of non-$L^1$ or
non-$L^2$ clustering. In the latter case one would again use the
\tit{Cauchy-Schwarz-inequality} (as in section 3.2). To illustrate the
method we choose the non-$L^1$ procedure.\\[0.3cm]
So let us assume
\begin{equation}W^T(y_1,\ldots,y_{l-1})\not\in L^1\end{equation}
We now choose a certain polynomial
\begin{equation}P(y):=(1+\sum
y_i^2)^{(l-1)\cdot\alpha/2}\end{equation}
with the (minimal) exponent $\alpha$ so chosen that
\begin{equation}F(y):=P(y)^{-1}\cdot W^T(y)\in L^1\end{equation}
On the other side, we define the fluctuation operators with the
exponent $\gamma$, which will be adjusted later
\begin{equation}A_R^F:=R^{-\gamma}\cdot A_R\end{equation}
It follows
\begin{equation}W^T(y)=P(y)\cdot F(y)\end{equation}
with $F(y)$ an $L^1$-function.
For the limit correlation functions we then get
\begin{equation}\langle A_R^F(1)\cdots A_R^F(l)\rangle^T=R^{ln}\cdot
R^{-l\gamma}\cdot\int \hat{F}(q)\cdot \hat{P}(q)\cdot
\hat{f}(Rp_1)\cdots \hat{f}(-Rq_{l-1})\prod dq_i\end{equation}
(cf. section 3.1)
\begin{bem} We write the Fourier transform of $P(y)$ formally as
\begin{equation}\hat{P}(q)=(1+\sum
D_{q_i}^2)^{(l-1)\cdot\alpha/2}\end{equation}
(with $D_{q_i}$ the partial derivations). For non-integer $(l-1)\cdot\alpha/2$ this is a \tit{pseudo-differential
operator}. Actually, we do not want to dwell too much on this ambitious
subject at the moment. What we in fact only need are the scaling
properties of the expression. If one wants to be careful one may
equally well take the explicit expression for the Fourier transform of
the above polynomial in the $y$-coordinates applied to the product of
the $f_R$'s and exploit its scaling properties.
\end{bem}
In any case, we get (with this proviso) and the usual variable
transformation $p_i':=Rp_i$:
\begin{equation}\langle A_R^F(1)\cdots
A_R^F(l)\rangle^T=R^{ln-l\gamma-(l-1)n+(l-1)\alpha}\cdot\int
\hat{F}(q'/R)\hat{P}(q')\cdot \hat{f}(p_1')\cdots
\hat{f}(-q_{l-1}')\prod dq_i'\end{equation}
Again only the explicit scaling prefactor matters in the limit
$R\to\infty$. To get a finite result for all correlation functions we
have to adjust the scaling parameter, $\gamma$, so that the exponents
vanish or are negative, that is, we want to have for $l=2$:
\begin{equation}n-2\gamma+\alpha=0\rightarrow \gamma=(n+\alpha)/2 \end{equation}
in order to get a non-trivial result. Inserting this $\gamma$ in the
general expression for $l\geq 3$, we get the condition
\begin{equation}2n-ln+(l-2)\alpha\leq 0\end{equation}
We know, on the other side, that $\alpha